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Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$.

Consider distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_{a,m}$ be distribution at every $m$.

$$\mbox{Case }(1)\mbox{: }a_{i+1}=a_i+\theta(\log^ka_i)$$ $$\mbox{Case }(2)\mbox{: }a_{i+1}=a_i+\theta(a_i^{\frac{1}{k}})$$ where $k$ is a positive constant.

$\mathsf{\underline{Conjecture}}$: I think at a fixed $k>0$, there will be constants $c_k>0, m(a,c_k)\in\Bbb N$ such that: $$\mbox{Case }1\mbox{: }\forall m>m(a,c_k)\mbox{, }H(\mathcal{P}_{a,m})>c_k$$ $$\mbox{Case }2\mbox{: }\forall m>0\mbox {, }H(\mathcal{P}_{a,m})<c_k.$$

Follow from relevant link Entropy difference dominance of sequences

Generalization is in here Limiting Entropy of deterministic sequences - 2

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$.

Consider distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_{a,m}$ be distribution at every $m$.

$$\mbox{Case }(1)\mbox{: }a_{i+1}=a_i+\theta(\log^ka_i)$$ $$\mbox{Case }(2)\mbox{: }a_{i+1}=a_i+\theta(a_i^{\frac{1}{k}})$$ where $k$ is a positive constant.

$\mathsf{\underline{Conjecture}}$: I think at a fixed $k>0$, there will be constants $c_k>0, m(a,c_k)\in\Bbb N$ such that: $$\mbox{Case }1\mbox{: }\forall m>m(a,c_k)\mbox{, }H(\mathcal{P}_{a,m})>c_k$$ $$\mbox{Case }2\mbox{: }\forall m>0\mbox {, }H(\mathcal{P}_{a,m})<c_k.$$

Follow from relevant link Entropy difference dominance of sequences

Generalization is in here Limiting Entropy of deterministic sequences - 2

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$.

Consider distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_{a,m}$ be distribution at every $m$.

$$\mbox{Case }(1)\mbox{: }a_{i+1}=a_i+\theta(\log^ka_i)$$ $$\mbox{Case }(2)\mbox{: }a_{i+1}=a_i+\theta(a_i^{\frac{1}{k}})$$ where $k$ is a positive constant.

$\mathsf{\underline{Conjecture}}$: I think at a fixed $k>0$, there is a constant $c_k$ such that in case $2,$ $$\forall m\mbox{, }H(\mathcal{P}_{a,m})<c_k$$ while there exists an $m(a,c_k)$ such that in case $1,$ $$\forall m>m(a,c_k)\mbox{, }H(\mathcal{P}_{a,m})>c_k.$$

Follow from relevant link Entropy difference dominance of sequences

Generalization is in here Limiting Entropy of deterministic sequences - 2

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$.

Consider distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_{a,m}$ be distribution at every $m$.

$$\mbox{Case }(1)\mbox{: }a_{i+1}=a_i+\theta(\log^ka_i)$$ $$\mbox{Case }(2)\mbox{: }a_{i+1}=a_i+\theta(a_i^{\frac{1}{k}})$$ where $k$ is a positive constant.

$\mathsf{\underline{Conjecture}}$: I think at a fixed $k>0$, there will be constants $c_k>0, m(a,c_k)\in\Bbb N$ such that: $$\mbox{Case }1\mbox{: }\forall m>m(a,c_k)\mbox{, }H(\mathcal{P}_{a,m})>c_k$$ $$\mbox{Case }2\mbox{: }\forall m>0\mbox {, }H(\mathcal{P}_{a,m})<c_k.$$

Follow from relevant link Entropy difference dominance of sequences

Generalization is in here Limiting Entropy of deterministic sequences - 2

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$.

Consider distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_{a,m}$ be distribution at every $m$.

$$\mbox{Case }(1)\mbox{: }a_{i+1}=a_i+\theta(\log^ka_i)$$ $$\mbox{Case }(2)\mbox{: }a_{i+1}=a_i+\theta(a_i^{\frac{1}{k}})$$ where $k$ is a positive constant.

$\mathsf{\underline{Conjecture}}$: I think at a fixed $k>0$, there is a constant $c_k$ such that in case $2,$ $$\forall m\mbox{, }H(\mathcal{P}_{a,m})<c_k$$ while there exists an $m(a,k)$ such that in case $1,$ $$\forall m>m(a,k)\mbox{, }H(\mathcal{P}_{a,m})>c_k.$$

Follow from relevant link Entropy difference dominance of sequences

Generalization is in here Limiting Entropy of deterministic sequences - 2

Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$.

Consider distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_{a,m}$ be distribution at every $m$.

$$\mbox{Case }(1)\mbox{: }a_{i+1}=a_i+\theta(\log^ka_i)$$ $$\mbox{Case }(2)\mbox{: }a_{i+1}=a_i+\theta(a_i^{\frac{1}{k}})$$ where $k$ is a positive constant.

$\mathsf{\underline{Conjecture}}$: I think at a fixed $k>0$, there is a constant $c_k$ such that in case $2,$ $$\forall m\mbox{, }H(\mathcal{P}_{a,m})<c_k$$ while there exists an $m(a,c_k)$ such that in case $1,$ $$\forall m>m(a,c_k)\mbox{, }H(\mathcal{P}_{a,m})>c_k.$$

Follow from relevant link Entropy difference dominance of sequences

Generalization is in here Limiting Entropy of deterministic sequences - 2